Let K₀ be an algebraic function field of one variable over a Hilbertian field F of positive characteristic p. Let K be a finite Galois extension of K₀. We prove that every finite embedding problem 1-->H-->G-->Gal(K/K₀)-->1 whose kernel H is a p-group is properly solvable.

Moreover, the solution can be chosen to locally coincide with finitely many, given in advance, weak local solutions. Finally, and this is the main point of this work, the number of prime divisors of K₀/F that ramify in the solution field is bounded by the number of prime divisors of K₀ that ramify in K plus the length of the maximal G-invariant sequence of subgroups of H.

The authors are indebted to Aharon Razon for carefully reading drafts of this work and for his useful comments. Special thanks go to the anonymous referee who suggested an essential simplification of an earlier version of the proof of Proposition 6.8.